Calculus Examples

$lim_{x \to 8} c \sqrt{4 x + 3} - \sqrt{2 x - 1}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$lim_{x \to 8} c \sqrt{4 x + 3} - lim_{x \to 8} \sqrt{2 x - 1}$

Step 2

Move the term $c$ outside of the limit because it is constant with respect to $x$ .

$c lim_{x \to 8} \sqrt{4 x + 3} - lim_{x \to 8} \sqrt{2 x - 1}$

Step 3

Move the limit under the radical sign.

$c \sqrt{lim_{x \to 8} 4 x + 3} - lim_{x \to 8} \sqrt{2 x - 1}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$c \sqrt{lim_{x \to 8} 4 x + lim_{x \to 8} 3} - lim_{x \to 8} \sqrt{2 x - 1}$

Step 5

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$c \sqrt{4 lim_{x \to 8} x + lim_{x \to 8} 3} - lim_{x \to 8} \sqrt{2 x - 1}$

Step 6

Evaluate the limit of $3$ which is constant as $x$ approaches $8$ .

$c \sqrt{4 lim_{x \to 8} x + 3} - lim_{x \to 8} \sqrt{2 x - 1}$

Step 7

Move the limit under the radical sign.

$c \sqrt{4 lim_{x \to 8} x + 3} - \sqrt{lim_{x \to 8} 2 x - 1}$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$c \sqrt{4 lim_{x \to 8} x + 3} - \sqrt{lim_{x \to 8} 2 x - lim_{x \to 8} 1}$

Step 9

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$c \sqrt{4 lim_{x \to 8} x + 3} - \sqrt{2 lim_{x \to 8} x - lim_{x \to 8} 1}$

Step 10

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$c \sqrt{4 lim_{x \to 8} x + 3} - \sqrt{2 lim_{x \to 8} x - 1 \cdot 1}$

Step 11

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 11.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$c \sqrt{4 \cdot 8 + 3} - \sqrt{2 lim_{x \to 8} x - 1 \cdot 1}$

Step 11.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$c \sqrt{4 \cdot 8 + 3} - \sqrt{2 \cdot 8 - 1 \cdot 1}$

Step 12

Simplify each term.

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Step 12.1

Multiply $4$ by $8$ .

$c \sqrt{32 + 3} - \sqrt{2 \cdot 8 - 1 \cdot 1}$

Step 12.2

Add $32$ and $3$ .

$c \sqrt{35} - \sqrt{2 \cdot 8 - 1 \cdot 1}$

Step 12.3

Multiply $2$ by $8$ .

$c \sqrt{35} - \sqrt{16 - 1 \cdot 1}$

Step 12.4

Multiply $- 1$ by $1$ .

$c \sqrt{35} - \sqrt{16 - 1}$

Step 12.5

Subtract $1$ from $16$ .

$c \sqrt{35} - \sqrt{15}$